July  2013, 33(7): 2621-2629. doi: 10.3934/dcds.2013.33.2621

The local $C^1$-density of stable ergodicity

1. 

College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China

Received  June 2012 Revised  November 2012 Published  January 2013

In this paper, we prove that stable ergodicity is $C^1$-dense among conservative partially hyperbolic systems which, in a stable way, have two ergodic measures such that one has all center Lyapunov exponents non-negative and the other one has all center Lyapunov exponents non-positive.
Citation: Yunhua Zhou. The local $C^1$-density of stable ergodicity. Discrete and Continuous Dynamical Systems, 2013, 33 (7) : 2621-2629. doi: 10.3934/dcds.2013.33.2621
References:
[1]

F. Abdenur, C. Bonatti, S. Crovisier, L. Diaz and L. Wen, Periodic points and homoclinic classes, Ergodic Theory Dynam. Systems, 27 (2007), 1-22. doi: 10.1017/S0143385706000538.

[2]

D. V. Anosov and Ya. Sinai, Certain smooth ergodic systems, UspehiMat. Nauk, 22 (1967), 107-172.

[3]

A. Ávila, On the regularization of conservative maps, Acta Math., 205 (2010), 5-18. doi: 10.1007/s11511-010-0050-y.

[4]

A. Ávila and J. Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Trans. Amer. Math. Soc., 364 (2012), 2883-2907. doi: 10.1090/S0002-9947-2012-05423-7.

[5]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic systems, Ann. of Math. (2), 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423.

[6]

C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104. doi: 10.1007/s00222-004-0368-1.

[7]

C. Bonatti and L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143 (1996), 357-396. doi: 10.2307/2118647.

[8]

C. Bonatti and L. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), 469-525. doi: 10.1017/S1474748008000030.

[9]

C. Bonatti, C. Matheus, M. Viana and A. Wilkinson, Abundance of stable ergodicity, Comment. Math. Helv., 79 (2004), 753-757. doi: 10.1007/s00014-004-0819-8.

[10]

K. Burns, D. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Stat. Phys., 108 (2002), 927-942. doi: 10.1023/A:1019779128351.

[11]

K. Burns, C. Pugh, M. Shub and A. Wilkinson, Recent results about stable ergodicity, in "Smooth ergodic theory and its applications (Seattle WA, 1999)", 69 of "Procs. Symp. Pure Math.", Amer. Math. Soc., (2001), 327-366.

[12]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math., 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451.

[13]

D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense, Astérisque, 287 (2003), 33-60.

[14]

M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2), 140 (1994), 295-329. doi: 10.2307/2118602.

[15]

C. Liang, W. Sun and J. Yang, Some results on perturbations to Lyapunov exponents, preprint, arXiv:1011.5299

[16]

C. Pugh and M. Shub, Stable ergodicity and partial hyperbolicity, in "1st International Conference on Dynamical Systems" (eds. F. Ledrappier etal.), Montevideo, Uruguay, (1995) - a tribute to Ricardo Mañé. Proceedings. Harlow: Longman. Pitman Res. Notes Math. Ser., 362 (1996), 182-187.

[17]

F. Rodríguez Hertz, M. A. Rodríguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381. doi: 10.1007/s00222-007-0100-z.

[18]

F. Rodríguez Hertz, M. A. Rodríguez Hertz, A. Tahzibi and R. Ures, Creation of blenders in the conservative setting, Nonlinearity, 23 (2010), 211-223. doi: 10.1088/0951-7715/23/2/001.

[19]

F. Rodríguez Hertz, M. A. Rodríguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and non-uniform hyperbolicity, Duke Math. J., 160 (2011), 599-629. doi: 10.1215/00127094-1444314.

show all references

References:
[1]

F. Abdenur, C. Bonatti, S. Crovisier, L. Diaz and L. Wen, Periodic points and homoclinic classes, Ergodic Theory Dynam. Systems, 27 (2007), 1-22. doi: 10.1017/S0143385706000538.

[2]

D. V. Anosov and Ya. Sinai, Certain smooth ergodic systems, UspehiMat. Nauk, 22 (1967), 107-172.

[3]

A. Ávila, On the regularization of conservative maps, Acta Math., 205 (2010), 5-18. doi: 10.1007/s11511-010-0050-y.

[4]

A. Ávila and J. Bochi, Nonuniform hyperbolicity, global dominated splittings and generic properties of volume-preserving diffeomorphisms, Trans. Amer. Math. Soc., 364 (2012), 2883-2907. doi: 10.1090/S0002-9947-2012-05423-7.

[5]

J. Bochi and M. Viana, The Lyapunov exponents of generic volume preserving and symplectic systems, Ann. of Math. (2), 161 (2005), 1423-1485. doi: 10.4007/annals.2005.161.1423.

[6]

C. Bonatti and S. Crovisier, Récurrence et généricité, Invent. Math., 158 (2004), 33-104. doi: 10.1007/s00222-004-0368-1.

[7]

C. Bonatti and L. Díaz, Persistent nonhyperbolic transitive diffeomorphisms, Ann. of Math., 143 (1996), 357-396. doi: 10.2307/2118647.

[8]

C. Bonatti and L. Díaz, Robust heterodimensional cycles and $C^1$-generic dynamics, J. Inst. Math. Jussieu, 7 (2008), 469-525. doi: 10.1017/S1474748008000030.

[9]

C. Bonatti, C. Matheus, M. Viana and A. Wilkinson, Abundance of stable ergodicity, Comment. Math. Helv., 79 (2004), 753-757. doi: 10.1007/s00014-004-0819-8.

[10]

K. Burns, D. Dolgopyat and Ya. Pesin, Partial hyperbolicity, Lyapunov exponents and stable ergodicity, J. Stat. Phys., 108 (2002), 927-942. doi: 10.1023/A:1019779128351.

[11]

K. Burns, C. Pugh, M. Shub and A. Wilkinson, Recent results about stable ergodicity, in "Smooth ergodic theory and its applications (Seattle WA, 1999)", 69 of "Procs. Symp. Pure Math.", Amer. Math. Soc., (2001), 327-366.

[12]

K. Burns and A. Wilkinson, On the ergodicity of partially hyperbolic systems, Ann. of Math., 171 (2010), 451-489. doi: 10.4007/annals.2010.171.451.

[13]

D. Dolgopyat and A. Wilkinson, Stable accessibility is $C^1$ dense, Astérisque, 287 (2003), 33-60.

[14]

M. Grayson, C. Pugh and M. Shub, Stably ergodic diffeomorphisms, Ann. of Math. (2), 140 (1994), 295-329. doi: 10.2307/2118602.

[15]

C. Liang, W. Sun and J. Yang, Some results on perturbations to Lyapunov exponents, preprint, arXiv:1011.5299

[16]

C. Pugh and M. Shub, Stable ergodicity and partial hyperbolicity, in "1st International Conference on Dynamical Systems" (eds. F. Ledrappier etal.), Montevideo, Uruguay, (1995) - a tribute to Ricardo Mañé. Proceedings. Harlow: Longman. Pitman Res. Notes Math. Ser., 362 (1996), 182-187.

[17]

F. Rodríguez Hertz, M. A. Rodríguez Hertz and R. Ures, Accessibility and stable ergodicity for partially hyperbolic diffeomorphisms with 1D-center bundle, Invent. Math., 172 (2008), 353-381. doi: 10.1007/s00222-007-0100-z.

[18]

F. Rodríguez Hertz, M. A. Rodríguez Hertz, A. Tahzibi and R. Ures, Creation of blenders in the conservative setting, Nonlinearity, 23 (2010), 211-223. doi: 10.1088/0951-7715/23/2/001.

[19]

F. Rodríguez Hertz, M. A. Rodríguez Hertz, A. Tahzibi and R. Ures, New criteria for ergodicity and non-uniform hyperbolicity, Duke Math. J., 160 (2011), 599-629. doi: 10.1215/00127094-1444314.

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